Double bifurcation for a cubic Kolmogorov model

Abstract

Our work focuses on the study of a kind of new bifurcation behavior about limit cycles (namely double bifurcation) for a class of cubic Kolmogorov model with a degenerate critical point (1, 1). The positive critical point (1, 1) is a 3-multiple nilpotent fine focus which can become a fine focus of 5th order. By analyzing the change in stability, we show (1, 1) can yield 4 limit cycles. Moreover, the high-order critical point (1, 1) can be broken into three lower-order critical points including a real critical points and two complex critical points under small parameters’ perturbation. Under the condition letting the multiplicity be same one, obtained real critical point (1, 1) can become a fine focus of 2nd order and yield two small limit cycles. In sum, 6 limit cycle can bifurcate near point (1, 1). This is a kind of new bifurcation behavior via double perturbation; in addition, in terms of the Hilbert number of cubic Kolmogorov model, our results are also new.